# Definition:Constructed Semantics/Instance 1

## Definition

Let $\LL_0$ be the language of propositional logic.

The constructed semantics $\mathscr C_1$ for $\LL_0$ is used for the following results:

### Structures

Define the structures of $\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows.

Let $\PP_0$ be the vocabulary of $\LL_0$.

Let a mapping $v: \PP_0 \to \set {1, 2}$ be given.

Next, regard the following as definitional abbreviations:

 $(1)$ $:$ Conjunction $\ds \mathbf A \land \mathbf B$ $\ds =_{\text{def} }$ $\ds \map \neg {\neg \mathbf A \lor \neg \mathbf B}$ $(2)$ $:$ Conditional $\ds \mathbf A \implies \mathbf B$ $\ds =_{\text{def} }$ $\ds \neg \mathbf A \lor \mathbf B$ $(3)$ $:$ Biconditional $\ds \mathbf A \iff \mathbf B$ $\ds =_{\text{def} }$ $\ds \paren {\mathbf A \implies \mathbf B} \land \paren {\mathbf B \implies \mathbf A}$

It only remains to define $\map v {\neg \phi}$ and $\map v {\phi \lor \psi}$ recursively, by:

 $\ds \map v {\neg \phi}$ $:=$ $\ds \begin{cases} 1 & : \text{if } \map v \phi = 2 \\ 2 & : \text{if } \map v \phi = 1 \end{cases}$ $\ds \map v {\phi \lor \psi}$ $:=$ $\ds \begin{cases} 1 & : \text{if } \map v \phi = \map v \psi = 1 \\ 2 & : \text{otherwise} \end{cases}$

### Validity

Define validity in $\mathscr C_1$ by declaring:

$\models_{\mathscr C_1} \phi$ if and only if $\map v \phi = 2$